Related papers: On generating the ring of matrix semi-invariants

We know that $\mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $\mathbb{Z}^m_n= \mathbb{Z}_n \times\mathbb{Z}_n\times...\times\mathbb{Z}_n$ be the Cartesian product of $m$ rings…

In this article we give an algorithm for determining the generators and relations for the rings of semi-invariant functions on irreducible components of representation spaces for gentle string algebras. These rings of semi-invariants turn…

We introduce two new invariants of a Noetherian (standard graded) local ring $(R, \mathfrak m)$ that measure the number of generators of certain kinds of reductions of $\mathfrak m,$ and we study their properties. Explicitly, we consider…

Let $\mathbb{F}_{q}$ be a finite field of characteristic $2$ and $O_2^+(\mathbb{F}_{q})$ be the $2$-dimensional orthogonal group of plus type over $\mathbb{F}_{q}$. Consider the standard representation $V$ of $O_2^+(\mathbb{F}_{q})$ and the…

In this paper, by using a combinatorial approach, we establish a new upper bound for the F-threshold $c^\mm(\mm)$ of determinantal rings generated by maximal minors. We prove that $c^\mm(\mm)$ coincides with the $a$-invariant in the case of…

We consider three types of rings of supersymmetric polynomials: polynomial ones $\Lambda_{m,n}$, partially polynomial $\Lambda_{m,n}^{+y}$ and Laurent supersymmetric rings $\Lambda_{m,n}^{\pm}$. For each type of rings we give their…

We show by adopting Schertz's argument with the Siegel-Ramachandra invariant that singular values of certain quotients of the $\Delta$-function generate ring class fields over imaginary quadratic fields.

In this paper, we study the vector invariants, ${\bf{F}}[m V_2]^{C_p}$, of the 2-dimensional indecomposable representation $V_2$ of the cylic group, $C_p$, of order $p$ over a field ${\bf{F}}$ of characteristic $p$. This ring of invariants…

It is a classical problem to compute a minimal set of invariant polynomial generating the invariant ring of a finite group as an algebra. We present here an algorithm for the computation of minimal generating sets in the non-modular case.…

The First Fundamental Theorem of Invariant Theory describes a minimal generating set of the invariant polynomial ring under the action of some group $G$. In this note we give an elementary and direct proof for the $\operatorname{GL}_2(K)$…

Let $V$ be an $n$-dimensional algebraic representation over an algebraically closed field $K$ of a group $G$. For $m > 0$, we study the invariant rings $K[V^{ m}]^G$ for the diagonal action of $G$ on $V^m$. In characteristic zero, a theorem…

Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order $|G|$ of a finite group $G$, then the polynomial invariants of $G$ are generated by polynomials of degrees at most $|G|$.…

We investigate the rings of semi-invariants for tame string algebras A(n) of non-polynomial growth. We are interested in dimension vectors of band modules. We use geometric technique related to the description of coordinate rings on…

Let $\mathbf{x}_{k \times p}$ be a $k \times p$ matrix of variables and let $\mathbb{F}[\mathbf{x}_{k \times p}]$ be the polynomial ring in these variables. Given two weak compositions $\alpha,\beta \models_0 n$ of lengths $\ell(\alpha) =…

We consider a finite permutation group acting naturally on a vector space $V$ over a field $\Bbbk$. A well known theorem of G\"obel asserts that the corresponding ring of invariants $\Bbbk[V]^G$ is generated by invariants of degree at most…

For a faithful linear representation $V$ of a finite group $G$ in coprime characteristic, we show that if the field Noether number $\beta_{\mathrm{field}}$ is the minimum $d$ such that the invariant polynomials of degree $\leq d$ generate…

Let $K$ be a field and let $\mathbb N = \{1,2, \dots \}$. Let $R_n=K[x_{ij} \mid 1\le i\le n, j\in \mathbb N]$ be the ring of polynomials in $x_{ij}$ $(1 \le i \le n, j \in \mathbb N)$ over $K$. Let $S_n = Sym (\{1,2, \ldots, n \})$ and…

Let $S$ be an additively idempotent semiring and $\mathbf{M}_n(S)$ be the semiring of all $n\times n$ matrices over $S$. We characterize the conditions of when the semiring $\mathbf{M}_n(S)$ is congruence-simple provided that the semiring…

This is a study of the sequences of Betti numbers of finitely generated modules over a complete intersection local ring, $R$. The subsequences $\{\beta^R_{i}(M)\}_{i\geq 0}$ with even, respectively, odd $i$ are known to be eventually given…

Let V be a normal affine variety over the real numbers R, and let S be a semi-algebraic subset of V(R). We study the subring B(S) of the coordinate ring of V consisting of the polynomials that are bounded on S. We introduce the notion of…